Some applications of Gröbner-Shirshov bases to Lie algebras

Pub Date : 2024-07-23 DOI:10.1016/j.jpaa.2024.107773
Luis Mendonça
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Abstract

We show that if a countably generated Lie algebra H does not contain isomorphic copies of certain finite-dimensional nilpotent Lie algebras A and B (satisfying some mild conditions), then H embeds into a quotient of AB that is at the same time hopfian and cohopfian. This is a Lie algebraic version of an embedding theorem proved by C. Miller and P. Schupp for groups. We also prove that any finitely presentable Lie algebra is the quotient of a finitely presented, centerless, residually nilpotent and SQ-universal Lie algebra of cohomological dimension at most 2 by an ideal that can be generated by two elements as a Lie subalgebra. This is reminiscent of the Rips construction in group theory. In both results we use the theory of Gröbner-Shirshov bases.

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格伯纳-希尔肖夫基在李代数中的一些应用
我们证明,如果一个可数生成的Lie代数不包含某些有限维零势Lie代数的同构副本,并且(满足一些温和的条件)嵌入到一个同时是hopfian和cohopfian的商中。这是米勒(C. Miller)和舒普(P. Schupp)为群证明的嵌入定理的李代数版本。我们还证明,任何有限呈现的李代数都是一个同调维数至多为 2 的有限呈现、无中心、残差零potent 和 SQ-universal 李代数的商,商是一个可以由两个元素生成的理想的李子代数。这让人想起群论中的里普斯构造。在这两个结果中,我们都使用了格罗布纳-希尔绍夫基理论。
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